basic microstructure-macroproperty calculations - t. i. zohdi

7/27/2019 Basic Microstructure-Macroproperty Calculations - T. I. Zohdi

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Basic Microstructure-Macroproperty

Calculations

T. I. Zohdi

1 Introduction

Many modern materials are comprised of an assembly of various microscalecomponents, typically ground up particulates or fibers in a binding matrix, which areformed into engineering geometries. The overall properties of such materials are theaggregate response of the assemblage of interacting components (Fig.1). The macro-scopic properties are tailored to the application, for example in structural engineeringapplications, by choosing a harder particulate phase that serves as a stiffening agent

for a ductile, easy to form, base matrix material. Experiments to determine the appro-priate combinations of particulate, fiber, and matrix phases are time-consuming andexpensive. Theoretical resultsare invaluable, from a qualitative point of view, but pro-vide limited quantitative information. Accordingly, microstructure-macropropertynumerical computation has evolved in this area of research for the last several years,andisconsideredamaturesubject,andanindispensibletoolforengineersandappliedscientists, when used in conjunction with experiments and guided by theory.

Microstructure-macroproperty methods, referred to by many different terms,such as homogenization, regularization, mean field theory, upscaling, etc.in various scientific communities, are used to compute effective properties of het-erogeneous materials. We will use these terms interchangeably in this chapter, butusually refer to them using the term micro-macro computation. The usual approachis to compute a constitutive relation between averages, relating volume averagedfield variables, resulting in effective properties. Thereafter, the effective propertiescan be used in a macroscopic analysis. The volume averaging takes place over astatistically representative sample of material, referred to in the literature as a repre-sentative volume element (RVE). The internal fields to be volumetrically averagedmust be computed by solving a series of boundary value problems with test load-

T. I. Zohdi (B)Department of Mechanical Engineering, University of California, 6117 Etcheverry Hall,Berkeley, CA94720-1740, USAe-mail: [email protected]

M. Kachanov and I. Sevostianov (eds.), Effective Properties of Heterogeneous Materials, 365Solid Mechanics and Its Applications 193, DOI: 10.1007/978-94-007-5715-8_5, Springer Science+Business Media Dordrecht 2013


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366 T. I. Zohdi

Fig. 1 An engineeringstructure comprised of amatrix binder and particulateadditives

SAMPLE

MATERIAL

ENGINEERING DEVICE

ings. There is a vast literature of methods, dating back to Maxwell [52, 53] andLord Rayleigh [65], to estimate the overall macroscopic properties of heterogeneous

materials. For an authoritative review of the general theory of random heterogeneousmedia, see Torquato [71] for more mathematical homogenization aspects, Jikov etal. [39] for more mathematical aspects, for solid-mechanics inclined accounts of thesubject, Hashin [28], Mura [58], Nemat-Nasser and Hori [59], Huet [34, 35], foranalyses of cracked media, Sevostianov et al. [69] and for computational aspects,Zohdi and Wriggers [80] and, recently, Ghosh [17], Ghosh and Dimiduk [18].

Our objective in this chapter is to provide some very basic concepts in this area,illustrated by a model problem involving linear elasticity, where the mechanicalproperties of microheterogeneous materials are characterized by a spatially variable

elasticity tensor IE. In order to characterize the (homogenized) effective macroscopicresponse of such materials, a relation between averages

= IE : (1)

is sought, where

def=

1

||

d , (2)

and where and are the stress and strain tensor fields within a statistically rep-resentative volume element (RVE) of volume ||. The quantity, IE, is known asthe effective property and is the elasticity tensor used in usual structural analyses.Similarly, one can describe other effective quantities such as conductivity or dif-fusivity, in virtually the same manner, relating other volumetrically averaged fieldvariables. However, for the sake of brevity, we restrict ourselves to linear elastostaticsproblems.

Computational methods for the calculation of effective properties attempt todirectly compute the relation between averages over a statistically representativesample of material. For asample to be statistically representative it must usuallycontain a large number of heterogeneities (Fig. 1) and, therefore, the computationsover the RVE are still extremely large, but are of reduced computational effort incomparison with a direct attack on the direct (entire engineering device simu-lation, which is virtually impossible) problem. As mentioned, classical analytical


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Basic Microstructure-Macroproperty Calculations 367

methods provide excellent qualitative information, however, they are lacking in termsof quantitive information. It is for this reason that computational approaches havebecome extremely popular. However, computational methods still require strongguidance from analytical methods (and vice-versa), which will be highlighted in this

chapter.In summary, readily available computational power has led simulation to aug-

ment theory and experimentation as an essential tool for the scientists and engineersin the twenty-first century. Explicitly stated, the goal of computational methods inmicromechanics is to determine relationships between the microstructure and the

macroscopic response of a material, by direct numerical computation over a sta-

tistically representative sample of material, guided by analytical methods, in order

to reduce laboratory time and expense for analysis and synthesis of new materials.

This chapter provides an introduction to basic homogenization theory and corre-

sponding computational methods, suitable for researchers in the applied sciences,mechanics and mathematics who have an interest in the analysis of new materials.It is assumed that readers have some familiarity with solid mechanics and the FiniteElement Method. This chapter draws heavily on the book Introduction to compu-tational micromechanics of Zohdi and Wriggers [80], and we refer the reader tothat document if they wish more extensive mathematical details and backgroundinformation. The outline of this chapter is as follows:

Basic micro-modeling concepts, specifically averaging theorems, micro-macroenergy relations and effective property bounds are presented, which serve as aguide to efficient computation.

Simple and efficient numerical procedures, based on the Finite Element Method,to simulate the response of samples of heterogeneous material, guided by theory,are outlined.

Numerical micro-macro examples are then given, and the results are interpreted. Some closing remarks are then provided on where the computational micro-macro

field is headed.

We remark that a field where computational micro-macro methods is of current

importance is in thermo-electromagnetic properties, which is well beyond the scopeof this chapter, but which is explored in depth in Zohdi [76].

2 Basic Micro-Macro Concepts

For a relation between averages to be useful it must be computed over a sample con-taining a statistically representative amount of material. This is a requirement that can

be formulated in a concise mathematical form. A commonly accepted macro/microcriterion used in effective property calculations is the so-called Hills condition, : = : . Hills condition [33] dictates the size requirements on the RVE.The classical argument is as follows. For any perfectly bonded heterogeneous body,


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368 T. I. Zohdi

in the absence of body forces, two physically important loading states satisfy Hillscondition. They are linear displacements of the form: (1) u| = E x = Eor (2) pure tractions in the form t| = L n = L; where E and L areconstant strain and stress tensors, respectively. Applying (1)- or (2)-type boundary

conditions to a large sample is a way of reproducing approximately what may beoccurring in a statistically representative microscopic sample of material in a macro-scopic body. Thus, thereis a clear interpretation to these testboundary conditions. Therequirement is that the sample must be large enough to have relatively small bound-

ary field fluctuations relative to its size and small enough relative to the macroscopic

engineering structure, forces us to choose boundary conditions that are uniform.

2.1 Testing Procedures

To determine IE, one specifies sixlinearly independent loadings of the form,

(1) u| = E(16) x or

(2) t| = L(16) n

where E(16) and L(16) are symmetric second order strain and stress tensors,with spatially constant (nonzero) components. This loading is applied to asample(such as in Fig. 1) of microheterogeneous material. Each independent loading yields

six different averaged stress components and hence provides six equations for theconstitutive constants in IE. In order for such an analysis to be valid, i.e. to makethe material data reliable, the sample of material must be small enough that it can beconsidered as a material point with respect to the size of the domain under analysis,but large enough to be a statistically representative sample of the microstructure.

If the effective response is assumed to be isotropic, then only one test loading(instead of usually six), containing non-zero dilatational ( tr3 and

tr3 ) and deviatoric

components(def= tr3 I and

def= tr3 I), are necessary to determinetheeffectivebulk and shear moduli:

3def=

tr3

tr3 and 2

def=

:

: . (3)

In general, in order to determine material properties of microheterogeneous material,one computes 36 constitutive constants1 Ei j k l in the following relation betweenaverages,

1 There are, of course, only 21 constants, since IE is symmetric.


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11223312

2313

=

E1111 E1122 E

1133 E

1112 E

1123 E

1113

E2211 E2222 E

2233 E

2212 E

2223 E

2213

E3311 E3322 E

3333 E

3312 E

3323 E

3313

E1211 E1222 E

1233 E

1212 E

1223 E

1213

E2311 E2322 E2333 E2312 E2323 E2313E1311 E

1322 E

1333 E

1312 E

1323 E

1313

112233

212

223213

. (4)

As mentioned before, each independent loading leads to six equations and hencein total 36 equations are generated by the independent loadings, which are used todetermine the tensor relation between average stress and strain, IE. IE is exactlywhat appears in engineering books as the property of a material. Theusualchoicesfor the six independent load cases are

E or L = 0 00 0 0

0 0 0

,0 0 00 00 0 0

,0 0 00 0 00 0

, 0 0 0 00 0 0

,0 0 00 0 0 0

, 0 0 0 0 0 0 0

,(5)

where is a load parameter. For completeness we record a few related fundamentalresults, which are useful in micro-macro mechanical analysis.

2.2 The Average Strain Theorem

If a heterogeneous body, see Fig. 2, has the following uniform loading on its surface:u| = E x, where E is a constant tensor then

=1

2||

(u + (u)T) d

=1

2||

1

(u + (u)T) d +

2

(u + (u)T) d

=1

2||

1

(u n + n u) dA +2

(u n + n u) dA

=1

2||

((E x) n + n (E x)) dA +

12

(|]u[| n + n |]u[|) dA

=1

2||

((E x) + (E x)T) d +

12

(|]u[| n + n |]u[|) dA

= E +1

2||

12

(|]u[| n + n |]u[|) dA, (6)

where (u ndef= ui nj ) is a tensor product of the vector u and vector n. |]u[| describes

the displacement jumps at the interfaces between 1 and 2. Therefore, only if thematerial is perfectly bonded, then = E. Note that the presence of finite body


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370 T. I. Zohdi

2

1

Fig. 2 Nomenclature for the averaging theorems

forcesdoesnotaffectthisresult.AlsonotethatthethirdlineinEq.6 isnotanoutcomeof the divergence theorem, but of a generalization that can be found in a variety ofplaces, for example Chandrasekharaiah and Debnath [4] or Malvern [51].

2.3 The Average Stress Theorem

Again we consider a body with t| = L n, where L is a constant tensor. We makeuse of the identity ( x) = ( ) x + x = f x + , where frepresents the body forces, and substitute this into the definition of the average stress

=1

||

( x) d +1

||

(f x) d

=1

||

( x) n dA +1

||

(f x) d

=1

|| (L x) n dA +1

|| (f x) d = L +1

|| (f x) d.(7)If there are no body forces,f = 0,then = L. Note that debonding (interfaceseparation) does not change this result.

2.4 Satisfaction of Hills Energy Condition

Consider a body with a perfectly bonded microstructure and f = 0, then

u

t dA =

u n dA =

(u ) d . With = 0, it follows that


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(u ) d =

u : d =

: d . If u| = E x and f = 0,

then

u t dA =

E x n dA =

(E x ) d (8)

=

(E x) : d = E : ||.

Noting that = E, we have

: = : . (9)

If t| = L n and f = 0, then u t dA =

u L n dA =

(u

L) d =

u : L d = L :

d . Therefore since = L, as before we

have : = : . Satisfaction of Hills condition guarantees that themicroscopic and macroscopic energy will be the same, and it implies the use of thetwo mentioned test boundary conditions on sufficiently large samples of material.

2.5 The Hill-Reuss-Voigt Bounds

Until recently, the direct computation of micromaterial responses was very difficult.Accordingly, classical approaches have sought to approximate or bound effectiveresponses. Many classical approaches start by splitting the stress field within a sampleinto a volume average, and a purely fluctuating part = + and we directlyobtain

0

: IE : d =

( : IE : 2 : IE : + : IE : ) d

= ( : IE : 2 : + : IE : )||= : (IE IE

) : ||. (10)

Similarly for the complementary case, with = + , and the followingassumption (microscopic energy equals the macroscopic energy)

: IE1 :

micro energy

= : IE1 :

macro energy

where = IE1 : (11)


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372 T. I. Zohdi

we have

0

: IE1 : d

=

( : IE1 : 2 : IE1 : + : IE

1 : ) d

= ( : IE1 : 2 : + : IE

1 : )||

= : (IE1 IE

1) : ||. (12)

Invoking Hills condition, whichis loading independent in this form, we have

IE11 Reuss IE IE Voigt

.(13)

This inequality means that the eigenvalues of the tensors IE IE11 andIE IE

are non-negative. The practical outcome of the analysis is that boundson effective properties are obtained. These bounds are commonly known as theHill-Reuss-Voigt bounds, for historical reasons. Voigt [73], in 1889, assumed thatthe strain field within a sample of aggregate of polycrystalline material, was uniform(constant),under uniform strain exterior loading. If the constant strain Voigt field isassumed within the RVE, = 0, then = IE : = IE : 0, which

implies IE

= IE . The dual assumption was made by Reuss [66], in 1929, whoapproximated the stress fields within the aggregate of polycrystalline material asuniform (constant), = 0, leading to = IE1 : = IE1 : 0, andthus IE = IE11 . Equality is attained in the above bounds ifthe Reuss or Voigtassumptions hold exactly, respectively.

Remark: Different boundary conditions (compared to the standard ones speci-fied earlier) are often used in computational homogenization analysis. For example,periodic boundary conditions are sometimes employed. Although periodicity con-ditions are really only appropriate for perfectly periodic media for many cases, it

has been shown that, in some cases, their use can more accurate effective responsesthan either linear displacement or uniform traction boundary conditions for a givensample size (for example, see Terada et al. [70] or Segurado and Llorca [67]). Peri-odic boundary conditions also satisfy Hills condition a priori. Another related typeof boundary conditions are so-called uniform-mixed types, whereby tractions areapplied on some parts of the boundary and displacements on other parts, generating,in some cases, effective properties that match those generated with uniform bound-ary conditions, but with smaller sample sizes (for example, see Hazanov and Huet[32]). Another approach is framing whereby the traction or displacement bound-

ary conditions are applied to a large sample of material, with the averaging beingcomputed on an interior subsample, to avoid possible boundary-layer effects. This issimilar to exploiting a St. Venant-type of effect, commonly used in solid mechanics,to avoid boundary layers. The approach provides a way of determining what the


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microstructure really experiences, without bias from the boundary loading. How-ever, generally, the advantages of one boundary condition over another diminishesas the sample increases in size.

2.6 Improved Estimates

Over the last half-century, improved estimates have been pursued, with a notablecontribution being the Hashin-Shtrikman bounds [2830]. The Hashin-Shtrikmanbounds are the tightest possible bounds on isotropic effective responses, gener-ated from isotropic microstructures, where the volumetric data and phase contrastsof the constituents are the only data known. For linearized elasticity applications,

for isotropic materials with isotropic effective (mechanical) responses, the Hashin-Shtrikman bounds (for a two-phase material) are as follows:

,def=1 +

v21

21+

3(1v2)31+41

2 +1 v2

112

+ 3v232+42

def=,+, (14)

and for the shear modulus

,def= 1 +

v21

21+

6(1v2)(1+21)

51(31+41)

2 +(1 v2)

1

12+

6v2(2+22)

52(32+42)

def= ,+,

(15)where 2 and 1 are the bulk moduli and 2 and 1 are the shear moduli of therespective phases (2 1 and 2 1), and where v2 is the second phase volumefraction. Such bounds are the tightest possible on isotropic effective responses, withisotropic two phase microstructures, where only the volume fractions and phasecontrasts of the constituents are known. Note that no geometric or distributionalinformation is required for the bounds.

Remark: There exist a multitude of other approaches which seek to estimate orbound the aggregate responses of microheterogeneous materials. A complete surveyis outside the scope of the present work. We refer the reader to the works of Hashin[28],Mura[58], Aboudi [1], Nemat-Nasser and Hori [59] and recently Torquato [71]for such reviews. Also, for in depth analyses, with extensions into nonlinear behavior,blending analytical, semi-analytical and numerical techniques, we refer the reader tothe extensive works of Llorca and co-workers: Segurado and Llorca [67], Gonzlezand Llorca [2224] Segurado et al. [68], Llorca [47, 48, 50], Poza and Llorca [63],Llorca and Gonzlez [49]. However, numerical methods have become the dominanttool to determining effective properties. In particular, Finite Element-based methodsare extremely popular, and we introduce the basics of this powerful tool, applied to

effective property calculations, in the next section.


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374 T. I. Zohdi

3 Computational/Statistical Testing Methods

Guided by the results of the previous section, we now investigate topics related to thenumerical simulation of the testing of mechanical responses of samples of micro-heterogeneous solid materials. The basis for this section follows from Zohdi andWriggers [80]. For illustration purposes only, we treat a model problem of cubicalsample of material filled with randomly distributed ellipsoidal particles in a homo-geneous matrix material. However, it should be clear to the reader that practicallyany microstructure can be analyzed with the presented approach. Three dimensionalnumerical examples employing the finite element method are given to illustrate theoverall analysis and computational testing process. The resulting microstructuresconsidered here are irregular and nonperiodic. A primary issue in the simulation ofsuch materials is the fact that only finite sized samples can be tested, leading not to

a single response, but a distribution of responses. This distribution of responses isthen interpreted employing potential energy principles.

3.1 A Boundary Value Formulation

We consider an isolated sample of heterogeneous material (Fig. 1), with domain ,under a given set of specified boundary loadings. In many problems of mathematical

physics the true solutions are nonsmooth, i.e. the strains and the stresses are notdifferentiable in the classical sense. For example in the equation of static equilibrium + f = 0, there is an implicit requirement that the stress was differentiable.2 Inmany applications, this is too strong of a requirement. Therefore, when solving suchproblems we have two options: (1) enforcement of jump conditions at every interfacewhere continuity is in question or (2) weak formulations (weakening the regularityrequirements). Weak forms, which are designed to accommodate irregular data andsolutions, are usually preferred. Numerical techniques employing weak forms, suchas the Finite Element Method, have been developed with the essential property that

whenever a smooth classical solution exists, it is also a solution to the weak formproblem. Therefore, we lose nothing by reformulating a problem in a weaker way.However, an important feature of such formulations is the ability to allow naturaland easy approximations to solutions in an energetic sense, which is desirable in theframework of mechanics.

2 Throughout this chapter, we consider only static linear elasticity, at infinitesimal strains, andspecialize approaches later for nonlinear and time dependent problems.


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3.2 Weak Formulations: The Foundation of Finite Element

Methods

Following Zohdi and Wriggers [80], to derive a direct weak form for a body, we takethe pointwise equilibrium equation ( + f, denoted the strong form) and forma scalar product with an arbitrary smooth vector valued function v, and integrateover the body,

( + f) v d =

r v d = 0, where r is called the

residual. We call v a test function. If we were to add a condition that we do thisfor all (

def= ) possible test functions then

( + f) v d =

r v d = 0,

v, implies r = 0. Therefore if every possible test function was considered, thenr = + f = 0 on any finite region in . Consequently, the weak and strongstatements would be equivalent provided the true solution is smooth enough to havea strong solution. Clearly, r can never be nonzero over any finite region in the body,because the test function will detect them.

Using the product rule of differentiation, ( v) = ( ) v + v : leads to, v,

( ( v) v : ) d +

f v d = 0, where we choose

the v from an admissible set, to be discussed momentarily. Using the divergencetheorem leads to, v,

v : d =

f v d +

n v dA, which leads to

v : d =

f v d + t v dA. If we decide to restrict our choices ofvs

to those such that v|u = 0, we have, where d is the applied boundary displacementon u , for infinitesimal strain linear elasticity

Find u, u|u

= d, such that v, v|u

= 0

v : IE : u d

def=B(u,v)

=

f v d +

t

t v dA def=F(v)

. (16)

This is called a weak form because it does not require the differentiability of thestress . In other words, the differentiability requirements have been weakened. It isclear that we are able to consider problems with quite irregular solutions. We observe

that if we test the solution with all possible test functions of sufficient smoothness,thentheweaksolutionisequivalenttothestrongsolution.We emphasize that providedthe true solution is smooth enough, the weak and strong forms are equivalent, which

can be seen by the above constructive derivation.

When we perform material tests satisfying Hills condition, we have, in the caseof displacement controlled tests (loading case (1)) u = and u| = E x orfor traction controlled tests (case (2)) t = and t| = L n. In either casewe consider f = 0 and that the material is perfectly bonded. We note that in case(1) Hills condition is satisfied with f = 0 (and no debonding) and in case (2) it is

satisfied even with debonding, however only iff=

0. The boundary value problemin Box 16 must be solved for each new sample, each possessing a different randommicrostructure (IE(x)). The solution is then post processed (averaged over the RVE)


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376 T. I. Zohdi

for the effective quantities. It is convenient to consider the RVE domain as a cube,and we shall do so for the remainder of the work.

3.3 Numerical Discretization

In order to computationally simulate effective responses, our choice for spatialdiscretization is the finite element method. There are essentially two choices to meshthe microstructure with the finite element method, a microstructure-nonconformingor a microstructure-conforming approach. We refer to a nonconforming approach asone which does not require the finite element boundaries to coincide with materialinterfaces when meshing the internal geometry (Fig. 3). This leads to material dis-

continuities within the finite elements. A conforming approach would impose thatthe element boundaries coincide with material interfaces and therefore the elementshavenomaterialdiscontinuitieswithinthem.Thereareadvantagesanddisadvantagesto both approaches. Nonconforming meshing has the advantage of rapid generationof structured internal meshes and consequently no finite element distortion arisingfrom the microstructure. This is critical to computational performance if iterativesolvers are to be used. The conforming meshing usually will require fewer finiteelements than the nonconforming approach for the same pointwise accuracy. How-ever, the disadvantages are the (extremely difficult) mesh generation for irregular

microstructures in three dimensions. Even if such microstructures can be meshed ina conforming manner, the finite element distortion leads to stiffness matrix ill con-ditioning and possible element instability (element nonconvexity). For numericalstudies comparing the meshing approaches, see Zohdi et al. [82]. Our emphasis is onstudying irregular microstructures, specifically randomly dispersed particulates, andrapidly evaluating them during the testing process. Therefore, we have adopted thenonconforming approach. Inherent in the nonconforming approach is the integrationof discontinuous integrands. The topology is not embedded into the finite element apriori, as it would be in a conforming approach, via isoparametric maps onto material

Fig. 3 Microstructure-nonconforming meshing withmaterial discontinuities withinan element

NODES

FINITE ELEMENT MESH

FINITE ELEMENT

GAUSS POINTS


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interfaces. To some extent, if the elements are much smaller than the particle lengthscales, the topology will be approximately captured. However, one can improve thisrepresentation (Fig. 3). Since the finite element method is an integral-based method,the quadrature rules can be increased in an element by element fashion to better

capture the geometry in elements with material discontinuities. Many studies by theauthor have indicated that for efficient implementation, a 2/5 rule should be used,whereby a 2 2 2 Gauss rule (two evaluation/integration points in each direc-tion) if there is no material discontinuity in the element, and a 5 5 5 rule (fiveevaluation/integration points in each direction) if there is a material discontinuity.We emphasize that this procedure is used simply to accurately integrate elementalquantities with discontinuities. For example in a series of numerical tests found inZohdi and Wriggers [78], the typical mesh density to deliver mesh insensitive results,for the quantities of interest in the upcoming simulations, was 9 9 9 trilinear

finite element hexahedra (approximately 22003000 degrees of freedom (DOF)) perparticle. For example, disk-type and a diamond-type microstructures, as resolved bythe meshing algorithm with a 24 24 24 trilinear hexahedra mesh density, witha total of 46875 degrees of freedom (approximately 9 9 9 hexahedra or 2344degrees of freedom per element), are shown in Fig. 4. In the sections that follow, weexplore these types of numerical tests in further detail.

Fig. 4 A random microstructure consisting of 20 non-intersecting particles. Left a diamond-typemicrostructure. Rightan oblate disk-type microstructure (aspect ratio of 3:1). Both microstructurescontain particles which occupy approximately 7% of the volume (Zohdi and Wriggers [80])


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378 T. I. Zohdi

3.4 Overall Testing Process: Numerical Examples

As considered before, a typical example of a composite material combination is thatof an aluminum matrix (77.9, 24.9 GPa) embedded with (stiffening) boron particles(230,172GPa).WechoseAluminum/Boronasamaterialcombinationwhichexhibitssignificant enough mismatch in the mechanical properties to be representative ofa wide range of cases. All tests were run on a single workstation. Such standardhardware is available in most academic and industrial work places, therefore suchsimulations are easily reproducible elsewhere for other parameter selections.

Successive Sample Enlargement

In a firstset of tests, the number of particles contained in a sample were increasedholding thevolume fractionconstant. During the tests, we repeatedly refined themeshto obtain mesh-invariant macroscopic responses. A sample/particle size ratio wasused as a microstructural control parameter. This was done by defining a subvolume

size Vdef= LLL

N, where N is the number of particles in the entire sample and where

L is the length of the (cubical L L L) sample. A generalized diameter (and radius)was defined, d = 2r, which was the diameter of the smallest sphere that can enclosea single particle, of possibly non-spherical shape (if desired). The ratio between

the generalized radius and the subvolume was defined by

def

=

r

V 13 . For a variety ofnumerical tests, discussed momentarily, the typical mesh density to deliver invariantvolumetrically averaged responses was 9 9 9 trilinear finite element hexahedra(approximately 22003000 degrees of freedom) per particle. We used = 0.375,which resulted in a (fixed) volume fraction of approximately 22 %. The followingparticle per sample sequence was used to study the dependence of the effectiveresponses on the sample size: 2 (5184 DOF), 4 (10125 DOF), 8 (20577 DOF), 16(41720 DOF), 32 (81000 DOF) and 64 (151959 DOF) particles. In order to obtainmore reliable response data for each particle number set, the tests were performed

five times (each time with a different particulate distribution) and the responsesaveraged. Throughout the tests, we considered a single combined boundary loadingsatisfying Hills condition, u| = E x, Ei j = 0.001, i, j = 1, 2, 3. We trackedthe strain energy, as well as and , as defined in Eq.(3). Table 1 and Fig. 5depict the dependency of the responses with growth in particle number. Justified bythe somewhat ad-hoc fact that for three successive enlargements of the number ofparticles, i.e. 16, 32 and 64 particle samples, the responses differed from one another,on average, by less than 1 %, we selected the 20-particle microstructures for furthertests. We remark that we applied a 2/5 rule, i.e. a 2 2 2 Gauss rule if there is

no discontinuity in the element, and a 5 5 5 rule if there is a discontinuity, whichis consistent with the earlier derivation in the work. The microstructure, as seen bythis mesh density, is shown in Fig. 6.


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Table 1 Results of successive sample enlargements

Particles dL

Num. unknowns (GPa) (GPa)

2 0.595 5184 98.2 46.74 0.472 10125 97.3 44.3

8 0.375 20577 96.5 43.216 0.298 41720 96.2 42.532 0.236 81000 95.9 41.664 0.188 151959 95.7 41.4

Five tests, each with a different random distribution, were performed at each sample/particulate sizeratio level to obtain somewhat representative data (Zohdi and Wriggers [80])

(64)(32)

(16)

(8)

(4)

10 20 4030 50 60

:( ):1/2

=

(GPa)

(2)

44

42

48

46

40

2

d/L=0.595

d/L= 0.472

d/L= 0.375

d/L=0.298

d/L= 0.236

d/L=0.188

Fig. 5 The values of the effective shear responses for samples containing increasingly largernumbers of particles. One hundred tests were performed per particle/sample combination and theresults averaged (Zohdi and Wriggers [80])

Multiple Sample Tests

For further tests, we simulated 100 different samples, each time with a differentrandom distribution of 20 nonintersecting particles occupying 22 % of the volume(= 0.375). Consistent with the previous tests mesh densities per particle, we useda 24 24 24 mesh ( 9 9 9 trilinear hexahedra or 2344 DOF per particle,46875 DOF per test sample). The plots of the behavior of the various quantities ofinterest are shown Fig.7. The averages, standard deviations and maximum-minimumof these quantities are tabulated in Table2. For the 100 sample tests, with 20 particlesper sample, the results for the effective responses were

91.37 = 1

1

= 96.17 = 111.79,

30.76 = 11 = 42.35 = 57.68,

(17)


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380 T. I. Zohdi

Fig. 6 Left a random microstructure consisting of 20 non-intersecting boron spheres, occupyingapproximately 22% of the volume in an aluminum matrix, as resolved with a 24 24 24 trilinearhexahedra mesh density for a total of 46875 degrees of freedom (approximately 9 9 9 hexahedraor 2344 degrees of freedom per element). A 2/5 rule, i.e. a 2 2 2 Gauss rule if there is nodiscontinuity in the element, and a 5 5 5 rule if there is a discontinuity, was used. Righta zoomon one particle (Zohdi and Wriggers [80])

EFFECTIVE BULK MODULUS (GPa)

SAMPLES

95.75 96 96.25 96.5

5

10

15

20

EFFECTIVE SHEAR MODULUS (GPa)

SAMPLES

41 41.5 42 42.5 43

0

5

10

15

20

Fig. 7 100 samples:Lefta histogram for the variations in the effective bulk responses, of a blockwith 20 randomly distributed Boron spheres embedded in an Aluminum matrix. Rightvariations inthe effective shear responses (Zohdi and Wriggers [80])

Table 2 Results of 100 material tests for randomly distributed particulate microstructures (20spheres) (Zohdi and Wriggers [80])

Quantity (GPa) Average Stan. dev. Max-min

96.171 0.2025 0.950 42.350 0.4798 2.250


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where and are the averaged effective responses from the 100 tests, and wherethe lower and upper bounds are, respectively, the classical Reuss [66] and Voigt [73]bounds.Wealsocomparedthecomputedresultstothewell-knownHashin-Shtrikmanbounds [29, 30] which are, strictly speaking, only applicable to asymptotic cases of

an infinite (sample length)/(particulate length) ratio and purely isotropic macroscopicresponses. The bounds were as follows:

94.32 = () = 96.17 (+) = 102.38,

35.43 = () = 42.35 (+) = 45.64,(18)

where 1, 1 and 2,2 are the bulk and shear moduli for the matrix and particlephases. Despite the fact that the bounds are technically inapplicable for finite sized

samples, the computed results did fall within them. The time to preprocess, solveand postprocess each 20 particle finite element test took no more than one minuteon a single laptop. Therefore, as before, 100 of such tests lasted approximately onehour.

Remark: Let us now increase the number of samples to 512 samples (of the samesize as before). The number 512 is not accidental, since it is a common number ofindependent processors in modern parallel processing machines. Table 3 illustratesthat the averaged results are virtually identical to the 100 sample tests for all thequantities. Testing more and more samples will not help obtain better average results.

However, despite practically the same average values, one can observe from the Fig.8that the 512 sample tests have a more Gaussian distribution, relative to the 100 sampletests,for the responses. However, for even more accurate average responses, we musttest larger samples of material. This is explored further in the next section.

3.5 Increasing Sample Size

Beyond a certain threshold, it is simply impossible to obtain any more information bytesting samples of a certain size, even when ensemble averaging over many differentsamples. Longer-range interactions need to be included, which can be achieved bytesting larger samples. Accordingly, we increase the number of particles per sampleeven further, from 20 to 40 then 60, each time performing the 100 tests procedure.

Table 3 Results of 512 material samples, each containing 20 randomly distributed spheres (Zohdiand Wriggers [80])

Quantity (GPa) Average Stan. dev. Max-min

96.169 0.1967 1.203 42.353 0.4647 3.207


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382 T. I. Zohdi

EFFECTIVE BULK MODULUS (GPa)

SAMPL

ES

96 96.5

10

20

30

40

50

60

70

80

90

100

110

120

EFFECTIVE SHEAR MODULUS (GPa)

SAMPL

ES

41 42 43

20

40

60

80

100

120

140

160

Fig. 8 512 samples:Lefta histogram for the variations in the effective bulk responses, of a block

with 20 randomly distributed Boron spheres embedded in an Aluminum matrix. Rightvariations inthe effective shear responses (Zohdi and Wriggers [80])

With this information one can then extrapolate to a (giant) sample limit. The resultsfor the 40 and 60 particle cases are shown in Table 4 for 22 % boron volume fraction.Using these results, along with the 20 particle per sample tests, we have the followingcurve fits

= 94.527 + 5.909 dL

, R2 = 0.986,

= 39.345 + 10.775 dL

, R2 = 0.986,(19)

where L is the sample size, d is the diameter of the particles. Thus as dL

0,we obtain estimates of = 94.527GPa and = 39.345GPa as the asymptoticenergy, effective bulk modulus, and effective shear modulus, respectively. Indeed,

judging from the degree of accuracy of the curve-fit (R2 = 1.0 is perfect) for and (regression values of R2 = 0.986) the relations are reliable. The monotonicallydecreasing character of the testing curves (effective property versus sample size) is

to be expected, and is explained in detail in Zohdi and Wriggers [80] using minimumenergy principles.

Table 4 Results of material tests for randomly distributed particulate microstructures for 100( = 0.375, approximately 22%) samples of 40 and 60 particles per sample (Zohdi and Wriggers[80])

Particles dL

Num. unknowns Quantity (GPa) Average Stan. dev. Max-min

40 0.2193 98304 95.7900 0.1413 0.6600

40 0.2193 98304

41.6407 0.3245 1.59060 0.1916 139968 95.6820 0.1197 0.621460 0.1916 139968 41.4621 0.2801 1.503


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3.6 A Minimum Principle Interpretation of the Results

One can interpret testing many samples of materials, each with boundary loadingu|

K

= E x as a construction of an approximate, kinematically-admissible (how-ever, not statically-admissible) solution for a super-sample comprised of gluingthe smaller samples together, and applying u|K = E x to the boundary of thesuper-sample. To illustrate this, consider the following process for a large sampleof material with u| = E x:

1. Step 1: Take the sample, and cut it into N pieces, = NK=1K. The piecesdo not have to be the same size or shape, although for illustration purposes it isconvenient to take a uniform (regular) partitioning (Fig. 9)

2. Step 2: Test each piece (solve the subdomain BVP) with the loading: u|K =

E x. The function uK is the solution to the BVP posed over subsample K3. Step 3: Defining the following

Kdef= IE

K : K , IEdef

=

NK=1

IE

K

|K|

||, (20)

where udef= u1|1 + u2|2 ...uN|N, one is guaranteed the following

IE1

1 IE

IE

IE . (21)

The effective material ordering in Eq. 21, has been derived by Huet [36] and gener-alizations to nonuniform loading were developed by Zohdi and Wriggers [78]. Theproofs are provided in Zohdi and Wriggers [80], utilizing classical energy minimiza-tion principles.

Remark: The same process can be done for traction test loading cases: t|K =L n. If we repeat the partitioning process for an applied (internal) traction set of

Fig. 9 Partitioning a sampleinto smaller samples or equiv-alently combining smallersamples into a larger sam-ple, producing an overallkinematically-admissible(however, not statically-admissible) solution

k


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384 T. I. Zohdi

tests, the results will bound the response of the very large sample from below. Itis relatively straightforward to show that (using complementary energy principles)if one applies traction tests to the boundary of samples, that lower bounds result,summarized as follows (Zohdi and Wriggers [80]):

IE11 (IE1

)1 averaging tr action tests

IE IE

averaging displacement tests

IE ,

(22)

where K = IE1K : K and IE

1 def=N

K=1 IE1K

|K|||

. We emphasize that

IE1 = IE1 is an assumption which may not be true for a finite sized sample.Therefore, in theory, under the RVE assumption, traction tests form lower bounds

on the effective responses. However, computationally, traction tests pose difficulties,which are as follows:

1. Numerically pure traction boundary data cause rigid motions (singular FEMstiffness matrices), however this can be circumvented by extracting the rigidbody modes (three translations and three rotations) a priori.

2. The FEM is a method based upon generating kinematically admissible solutions. The traction tests result is based upon the assumption that statically admissibletrial field are generated. Statically admissible fields cannot be achieved by astandard FEM approach.

The derived results allow one to bound, above and below, the unknown RVE responsein terms of the ensemble averages. Related forms of the bounds have been derivedin various forms dating back to Huet [3438] Hazanov and Huet [32], Hazanov andAmieur [31] and Zohdi et al. [81, 83], Oden and Zohdi [61], Zohdi and Wriggers[7779], and Zohdi [75].

4 Summary and Closing Comments

The results derived here can be used in conjunction with a variety of methods toperformlarge-scalemicro-macro multilevelsimulations. For reviews see Zohdi andWriggers [80] and, recently, Ghosh [17], Ghosh and Dimiduk [18]. Noteworthy are

Multiscale methods: Fish and Wagiman [12], Fish et al. [1416] Fish and Belsky[68], Fish and Shek [11], Fish and Ghouli [10], Fish and Yu [13], Fish and Chen[9], Chen and Fish [5], Wentorf et al. [74], Ladeveze et al. [43], Ladeveze andDureisseix [41, 42], Ladeveze [40] and Champaney et al. [3],

Voronoi cell methods: Ghosh and Mukhopadhyay [21], Ghosh and Moorthy [19,20] Ghosh et al. [2527], Lee et al. [45], Li et al. [46], Moorthy and Ghosh [56]and Raghavan et al. [64],

Transformation methods: Moulinec et al. [57] and Michel et al. [54],


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Partitioning methods:Huet[3438],HazanovandHuet[32], Hazanov and Amieur[31] and

Adaptive hierarchical modeling methods: Zohdi et al. [81], Oden and Zohdi [61],Moes et al. [55], Oden and Vemaganti [60], Oden et al. [62] and Vemaganti and

Oden [72].

Particularly attractive are iterative domain decomposition type strategies, wherebya global domain is divided into nonoverlapping subdomains. On the interior sub-domain partitions an approximate globally kinematically admissible displacementis projected. This allows the subdomains to be mutually decoupled, and thereforeseparately solvable. The subdomain boundary value problems are solved with theexact microstructural representation contained within their respective boundaries,but with approximate displacement boundary data. The resulting microstructural

solution is the assembly of the subdomain solutions, each restricted to its corre-sponding subdomain. As in the ensemble testing, the approximate solution is farless expensive to compute than the direct problem. Numerical and theoretical stud-ies of such approaches have been studied by Huet [34], Hazanov and Huet [31],Zohdi et al. [81], Oden and Zohdi [61], Zohdi and Wriggers [7779], Zohdi [75]and Zohdi et al. [83]. Clearly, when decomposing the structure by a projection ofa kinematically admissible function onto the partitioning interfaces, regardless ofthe constitutive law, the error is due to the jumps in tractions at the interfaces (sta-tical inadmissibility). If the interfaces were in equilibrium, then there would be notraction jumps. Therefore, if the resulting approximate solution is deemed not accu-rate enough, via a-posteriori error estimation techniques, the decoupling function onthe boundaries of the subdomain is updated using information from the previouslycomputed solution, and the subdomains are solved again. Methods for updating sub-domain boundaries can be found in Zohdi et al. [83]. They bear a strong relation toalternating Schwarz methods (see Le Tallec [44] for reviews) and methods of equili-bration (see Ainsworth and Oden [2]). For more details, we refer the reader to Zohdiand Wriggers [80].

In closing, we mention that recently, several applications, primarily driven bymicro-technology, have emerged where the use of materials with tailored electro-

magnetic (dielectric) properties are necessary for a successful overall design. Thetailored aggregate properties are achieved by combining an easily moldable basematrix with particles having dielectric properties that are chosen to deliver (desired)effective properties. In many cases, the analysis of such materials requires the sim-ulation of the macroscopic and microscopic electromagnetic response, as well as itsresulting coupled thermal response, which can be important to determine possiblefailure in hot spots. This necessitates a thermo-mechanical stress analysis. We referinterested readers to Zohdi [76] for more details, on this emerging subject involvingmultiphysics.


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386 T. I. Zohdi

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